1Introduction to Partial Differential Equations
1277.1 Fundamental Solutions
21.1 What are Partial Differential Equations?
1287.2 Green’s Functions for PDEs
31.2 Classification of PDEs
1297.3 Green’s Identities
41.2.1 Order of PDEs
130First Green’s Identity:: Second Green’s Identity:
51.2.2 Linearity of PDEs
1317.4 Boundary Value Problems: Mathematical Question and Solution:
61.2.3 Number of Independent Variables
1327.5 Inhomogeneous PDEs: Mathematical Question and Solution:
71.3 Applications of PDEs
1337.6 Applications
81.4 Boundary and Initial Conditions
1341. Electrostatics and Magnetostatics:
91.4.1 Boundary Conditions: 1.4.2 Initial Conditions
1352. Heat Transfer:
101.5 Well-Posed Problems
1363. Fluid Dynamics:
111.6 Analytical and Numerical Solutions
1374. Quantum Mechanics:
121.6.1 Analytical Solutions
1385. Elasticity and Structural Analysis:
131.6.2 Numerical Solutions
1396. Electromagnetic Wave Propagation:
141.6.3 Advantages and Limitations of Analytical and Numerical Solutions
1407. Integral Equation Formulations:
15Advantages of Analytical Solutions:
1418. Inverse Problems:
16Limitations of Analytical Solutions:
142Conclusion
17Advantages of Numerical Solutions:
143Numerical Methods for PDEs
18Limitations of Numerical Solutions:
1448.1 Finite Difference Methods
19Conclusion
145Finite Difference Approximations:
20First-Order PDEs
146Finite Difference Schemes:: Stability, Consistency, and Convergence:
212.1 Linear First-Order PDEs
1478.2 Finite Element Methods
222.1.1 Classification of Linear First-Order PDEs: 2.1.2 Solution Methods for Linear First-Order PDEs
1488.3 Spectral Methods: Mathematical Question and Solution:
232.2 Nonlinear First-Order PDEs
1498.4 Stability and Convergence
242.2.1 Classification of Nonlinear First-Order PDEs: 2.2.2 Solution Methods for Nonlinear First-Order PDEs
150Stability:
252.3 Method of Characteristics
151Convergence:: Mathematical Question and Solution:
262.3.1 Concept of Characteristic Curves
1528.5 Error Analysis
272.3.2 Steps in the Method of Characteristics
153Types of Errors:
282.3.3 Advantages and Limitations
154Error Estimation and Control:: Mathematical Question and Solution:
292.3.4 Numerical Methods for First-Order PDEs
1558.6 Software Packages
302.4 Cauchy Problem
156Conclusion
312.4.1 Formulation of the Cauchy Problem
1571. Finite Difference Methods:
322.4.2 Cauchy Problem for Linear and Nonlinear PDEs
1582. Finite Element Methods:
33Linear First-Order PDEs:: Nonlinear First-Order PDEs:
1593. Spectral Methods:
342.4.3 Examples and Mathematical Questions
1604. Stability and Convergence:
352.4.4 Ill-Posed Cauchy Problems and Regularization
1615. Error Analysis:: 6. Software Packages:
362.5 Applications of First-Order PDEs
162Nonlinear PDEs
372.5.1 Traffic Flow Models
1639.1 Burgers’ Equation: Numerical Solution of Burgers’ Equation using Finite Differences:
382.5.2 Wave Propagation
1649.2 Korteweg-de Vries Equation
392.5.3 Geometric Optics
1651. Soliton Solution:
402.5.4 Other Applications
1662. Periodic Solution:: Numerical Solution of the KdV Equation:
412.6 Numerical Methods for First-Order PDEs
1679.3 Sine-Gordon Equation
422.6.1 Finite Difference Methods (FDM)
168Analytical Solutions to the Sine-Gordon Equation:
432.6.2 Finite Volume Methods (FVM)
1691. Soliton Solution:
442.6.3 Finite Element Methods (FEM)
1702. Periodic Solution:: Numerical Solution of the Sine-Gordon Equation:
452.6.4 Other Numerical Methods
1719.4 Shock Waves: Numerical Simulation of Shock Waves:
46Second-Order Linear PDEs
1729.5 Solitons: Numerical Simulation of Solitons:
473.1 Classification of Second-Order Linear PDEs
1739.6 Applications of Nonlinear PDEs
48Elliptic PDEs:
1741. Fluid dynamics:
49Parabolic PDEs:
1752. Nonlinear optics:
50Hyperbolic PDEs:: Practice Question:
1764. Condensed matter physics:
513.2 Separation of Variables: Practice Question:
1775. Quantum field theory:: 6. Astrophysics and cosmology:
523.3 Heat Equation
178Conclusion
53Analytical Solutions:
179Calculus of Variations
54Numerical Solutions:
18010.1 Functionals and Euler-Lagrange Equations
55Practice Question:: Mathematical Questions and Solutions:
18110.2 Variational Problems
563.4 Wave Equation
18210.3 Geodesics and Minimal Surfaces
57Analytical Solutions:
183Geodesics: Minimal Surfaces
58Numerical Solutions:: Practice Question:
18410.4 Isoperimetric Problems
593.5 Laplace’s Equation
18510.5 Ritz Method
60Analytical Solutions:
18610.6 Applications to PDEs
61Numerical Solutions:: Practice Question:
1871. Boundary Value Problems:
623.6 Boundary Value Problems
1882. Variational Principles:
63Analytical Solutions:
1893. Optimal Control Problems:: 4. Isoperimetric Problems:
64Numerical Solutions:: Practice Question:
190Conclusion
65Conclusion
191Elliptic PDEs
66Fourier Series
19211.1 Laplace’s Equation
674.1 Periodic Functions: Practice Question:
19311.2 Poisson’s Equation
684.2 Fourier Series Representation: Practice Question:
19411.3 Maximum Principles
694.3 Convergence of Fourier Series
19511.4 Dirichlet and Neumann Problems
70Dirichlet’s Conditions:
196The Dirichlet Problem:: The Neumann Problem:
71Convergence of Fourier Series for Discontinuous Functions:
19711.5 Green’s Functions
72Practice Question:: Mathematical Questions and Solutions:
19811.6 Applications
734.4 Applications of Fourier Series
1991. Electrostatics:
741. Signal Processing:
2002. Heat Conduction:
752. Communication Systems:
2013. Quantum Mechanics:: 4. Fluid Mechanics:
763. Control Systems:
202Conclusion
774. Electrical Engineering:
203Parabolic PDEs
785. Mechanical Engineering:
20412.1 Heat Equation
796. Quantum Mechanics:
20512.2 Diffusion Equations
807. Image and Signal Processing:
20612.3 Maximum Principles: Mathematical Questions and Solutions:
818. Numerical Analysis:: Practice Question:
20712.4 Initial and Boundary Value Problems
824.5 Fourier Integrals and Transforms
208Initial Conditions:: Boundary Conditions:
83Fourier Integrals:
20912.5 Finite Difference Methods
84Fourier Transforms:
21012.6 Applications
851. Signal Processing:
2111. Heat Transfer:
862. Image Processing:
2122. Diffusion Processes:
873. Partial Differential Equations:
2133. Finance and Economics:
884. Quantum Mechanics:
2144. Image Processing:
895. Spectroscopy:: Practice Question:
2155. Fluid Dynamics:
904.6 Generalized Fourier Series
2166. Population Dynamics:
911. Legendre Polynomials:
2177. Quantum Mechanics:
922. Bessel Functions:
2188. Materials Science:: Mathematical Questions and Solutions:
933. Spherical Harmonics:
219Conclusion
944. Chebyshev Polynomials:
220Hyperbolic PDEs
95Practice Question:: Mathematical Questions and Solutions:
22113.1 Wave Equation
96Separation of Variables
222Physical Interpretation:
975.1 Cartesian Coordinates
223Initial and Boundary Conditions:
98Mathematical Question 1:: Mathematical Question 2:
224Initial Conditions:
995.2 Cylindrical Coordinates
225Boundary Conditions (for a string of length L):: Mathematical Questions and Solutions:
100Applying the boundary conditions:
22613.2 Characteristics
101Mathematical Question 1:: Mathematical Question 2:
227Definition of Characteristics:
1025.3 Spherical Coordinates: Mathematical Question 1:
228Properties of Characteristics:: Mathematical Questions and Solutions:
1035.4 Sturm-Liouville Problems
22913.3 Domain of Dependence and Influence
104Example: Consider the Sturm-Liouville problem:
230Domain of Dependence:
105Properties of Sturm-Liouville Problems:
231Domain of Influence:
106Mathematical Question 1:
232Significance and Applications:: Mathematical Questions and Solutions:
107Mathematical Question 2:: Consider the Sturm-Liouville problem:
23313.4 Initial and Boundary Value Problems
1085.5 Eigenvalue Problems: Properties of Eigenvalue Problems:
234Initial Conditions:
109Conclusion
235Boundary Conditions:: Mathematical Questions and Solutions:
110Integral Transforms
23613.5 Finite Difference Methods
1116.1 Fourier Transforms
237Finite Difference Approximations:
1126.1.1 Fourier Series
238Explicit and Implicit Schemes:
1136.1.2 Fourier Transform
2391. Explicit Schemes:
1146.1.3 Fourier Transform Methods for PDEs: Mathematical Question 1:
2402. Implicit Schemes:
1156.2 Laplace Transforms
241Stability and Convergence:: Mathematical Questions and Solutions:
1166.2.1 Definition and Properties
24213.6 Applications
1176.2.2 Laplace Transform Methods for PDEs: Mathematical Question 1:
2431. Acoustics and Sound Waves:
1186.3 Inverse Transforms
2442. Electromagnetic Waves:
1196.3.1 Inverse Fourier Transform
2453. Seismology and Earthquake Waves:
1206.3.2 Inverse Laplace Transform
2464. Fluid Dynamics:
121Mathematical Question 1:: Mathematical Question 2:
2475. Quantum Mechanics:
1226.4 Convolution Theorem
2486. Numerical Simulations:
1236.5 Applications to PDEs
2497. Signal Processing:: Mathematical Questions and Solutions:
1246.6 Numerical Inversion Techniques
250Conclusion
125Conclusion
251Glossary
126Green’s Functions
252Index
